What You Should Learn Evaluate trigonometric functions of any angle Evaluate trigonometric functions of real numbers

Positive Trig Function Values STAR ALL Sine and its reciprocal are positive All functions are positive r r y y -x x -y -y r r TRIG CLASS Tangent and its reciprocal are positive Cosine and its reciprocal are positive

Positive, Negative or Zero? sin 240° Negative cos 300o Positive Positive tan 225o

Determine the Quadrant In which quadrant is θ if cos θ and tan θ have the same sign? Quadrants I and II

Determine the Quadrant In which quadrant is θ if cos θ is negative and sin θ is positive? Quadrant II

Determine the Quadrant In which quadrant is θ if cot θ and sec θ have opposite signs? Quadrants III and IV

Evaluating Trigonometric Functions for any Angle

Introduction Following is the definition of trigonometric functions of Any Angle.

Introduction Because r = cannot be zero, it follows that the sine and cosine functions are defined for any real value of  . However, when x = 0, the tangent and secant of  are undefined. For example, the tangent of 90  is undefined. Similarly, when y = 0, the cotangent and cosecant of  are undefined.

Example 1 – Evaluating Trigonometric Functions Let (–3, 4) be a point on the terminal side of  (see Figure 4.34). Find the sine, cosine, and tangent of . Figure 4.34

Example 1 – Solution Referring to Figure 4.34, you can see that x = –3, y = 4, and

Example 1 – Solution So, you have and cont’d

Introduction The signs of the trigonometric functions in the four quadrants can be determined easily from the definitions of the functions. For instance, because it follows that cos  is positive wherever x > 0, which is in Quadrants I and IV.

Using the Sign If and lies in Quadrant III, find sin and tan θ θ θ -1 -√3 2

Review Find the exact values of the other five trig functions for an angle θ in standard position, given 12 360o θ -5 13 270o

Your Turn: Let θ be an angle in Quadrant III such that sin θ = −5/13. Find a) sec θ and b) tan θ.  

Right Triangle Trigonometry Trigonometry is based upon ratios of the sides of right triangles. The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. θ hyp opp adj The sides of the right triangle are:  the side opposite the acute angle ,  the side adjacent to the acute angle ,  and the hypotenuse of the right triangle.

Right Triangle Trigonometry The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90o. A

S O H C A H T O A Trigonometric Ratios hyp opp θ adj The trigonometric functions are: sine, cosine, tangent, cotangent, secant, and cosecant. sin Ɵ = cos Ɵ = tan Ɵ = csc Ɵ = sec Ɵ = cot Ɵ = opp hyp adj S O H C A H T O A

Reciprocal Functions sin  = 1/csc  csc  = 1/sin  cos  = 1/sec  sec  = 1/cos  tan  = 1/cot  cot  = 1/tan 

Finding the ratios The simplest form of question is finding the decimal value of the ratio of a given angle. Find using calculator: sin 30 = sin 30 = cos 23 = tan 78 = tan 27 = sin 68 =

Using ratios to find angles It can also be used in reverse, finding an angle from a ratio. To do this we use the sin-1, cos-1 and tan-1 function keys. Example: sin x = 0.1115 find angle x. sin-1 0.1115 = shift sin ( ) x = sin-1 (0.1115) x = 6.4o 2. cos x = 0.8988 find angle x cos-1 0.8988 = shift cos ( ) x = cos-1 (0.8988) x = 26o

Calculate the trigonometric functions for  . 4 3 5  The six trig ratios are sin  = sin α = cos  = cos α = tan  = What is the relationship of α and θ? tan α = cot  = cot α = sec  = They are complementary (α = 90 – θ) sec α = csc  = csc α =

Cofunctions sin  = cos (90  ) cos  = sin (90  ) tan  = cot (90  ) cot  = tan (90  ) tan  = cot (π/2  ) cot  = tan (π/2  ) sec  = csc (90  ) csc  = sec (90  ) sec  = csc (π/2  ) csc  = sec (π/2  )

Finding an angle from a triangle To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle. We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio. 14 cm 6 cm C 1. Find angle C Identify/label the names of the sides. b) Choose the ratio that contains BOTH of the letters.

14 cm 6 cm C 1. We have been given the adjacent and hypotenuse so we use COSINE: Cos A = h a Cos A = Cos C = Cos C = 0.4286 C = cos-1 (0.4286) C = 64.6o

Find angle x 2. 8 cm 3 cm x Given adj and opp need to use tan: Tan A = a o Tan A = Tan x = Tan x = 2.6667 x = tan-1 (2.6667) x = 69.4o

3. 12 cm 10 cm y Given opp and hyp need to use sin: Sin A = sin A = sin x = sin x = 0.8333 x = sin-1 (0.8333) x = 56.4o

Finding a side from a triangle 7 cm k 30o 4. We have been given the adj and hyp so we use COSINE: Cos A = Cos A = Cos 30 = Cos 30 x 7 = k 6.1 cm = k

4 cm r 50o 5. We have been given the opp and adj so we use TAN: Tan A = Tan A = Tan 50 = Tan 50 x 4 = r 4.8 cm = r

12 cm k 25o 6. We have been given the opp and hyp so we use SINE: Sin A = sin A = sin 25 = Sin 25 x 12 = k 5.1 cm = k

x = x 5 cm 30o 1. Cos A = Cos 30 = x = 5.8 cm 4 cm r 50o 2. Tan 50 x 4 = r 4.8 cm = r Tan A = Tan 50 = 3. 12 cm 10 cm y y = sin-1 (0.8333) y = 56.4o sin A = sin y = sin y = 0.8333

Example: Given sec  = 4, find the values of the other five trigonometric functions of  . Solution: Draw a right triangle with an angle  such that 4 = sec  = = . adj hyp θ 4 1 Use the Pythagorean Theorem to solve for the third side of the triangle. sin  = csc  = = cos  = sec  = = 4 tan  = = cot  =

Applications Involving Right Triangles A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3. How tall is the Washington Monument? Solution: where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3  115(4.82882)  555 feet.

Fundamental Trigonometric Identities Reciprocal Identities sin  = 1/csc  cos  = 1/sec  tan  = 1/cot  cot  = 1/tan  sec  = 1/cos  csc  = 1/sin  Co function Identities sin  = cos(90  ) cos  = sin(90  ) sin  = cos (π/2  ) cos  = sin (π/2  ) tan  = cot(90  ) cot  = tan(90  ) tan  = cot (π/2  ) cot  = tan (π/2  ) sec  = csc(90  ) csc  = sec(90  ) sec  = csc (π/2  ) csc  = sec (π/2  ) Quotient Identities tan  = sin  /cos  cot  = cos  /sin  Pythagorean Identities sin2  + cos2  = 1 tan2  + 1 = sec2  cot2  + 1 = csc2 